If a Muslim realizes that granting a copy of the Qur’an for a non-Muslims as a present will help spread da`wah, then there is nothing wrong in that.

In this regard, Dr. Sano Koutoub Moustapha, professor of Fiqh and Its Principles, International Islamic University: regarding giving a copy of the Qur’an itself to an unbeliever, I would prefer not to do so unless there is a hope that he or she might benefit from it. In other words, if this would lead him or her to convert to Islam, I see no harm to do so. This is based on the legal maxim which says, necessities render an unlawful deed a lawful one.
Surely, you should not give it or distribute it to those people who might damage it or abuse it intentionally. Our beloved Prophet (peace and blessings be upon him) sent messages to unbelievers and those messages contained many verses from the Qur’an. In addition to that, for the sake of da`wah, one should do his best to introduce this Glorious Book to unbelievers with the hope that they convert to Islam later.
Surely, the prohibition stated in the Qur’anic verses that mean
(And lo! that verily is a tremendous oath, if ye but knew that (this) is indeed a noble Qur’an in a Book kept hidden which none toucheth save the purified)(Al-Waqi`ah 56: 76-79) does not mean non-Muslims but the devils, as a response to the accusation in the verse meaning (The devils did not bring it down)(Ash-Shu`araa’ 26: 210).”